Editing Electromagnetic four-potential (section)

== Gauge freedom ==

When [[Musical isomorphism|flattened]] to a [[one-form]] (in tensor notation, <math>A_\mu</math>), the four-potential <math>A</math> (normally written as a vector or, <math>A^\mu</math> in tensor notation) can be decomposed{{clarify|date=November 2022|reason=What are the operators? d is the four gradient, but what is delta. Which differential complex?}} via the [[Hodge theory|Hodge decomposition theorem]] as the sum of an [[Closed and exact differential forms|exact]], a coexact, and a harmonic form,
: <math>A = d \alpha + \delta \beta + \gamma</math>.

There is [[gauge freedom]] in {{math|''A''}} in that of the three forms in this decomposition, only the coexact form has any effect on the [[electromagnetic tensor]]
: <math>F = d A</math>.

Exact forms are closed, as are harmonic forms over an appropriate domain, so <math>d d \alpha = 0</math> and <math>d\gamma = 0</math>, always. So regardless of what <math>\alpha</math> and <math>\gamma</math> are, we are left with simply
: <math>F = d \delta \beta</math>.

In infinite flat Minkowski space, every closed form is exact. Therefore the <math>\gamma</math> term vanishes. Every gauge transform of <math>A</math> can thus be written as
: <math>A \Rightarrow A + d\alpha</math>.